Thus \(\bar{\mathcal {I}}_n\)  is simply the EDP-functional for jump processes of [1]}. The works [2]}, [3]}, [4]} contain an extensive overview and discussion on how \(\bar{\mathcal {I}}_n\)  is the expected rate functional for a large-deviation principle for the empirical measures of independent jump processes, how the reversibility of the process ensures a possible splitting in both the interacting and non-interacting case, and how for complex-balanced systems this can even be done in the irreversible setting. Moreover, for an implicit decomposition using measure-dependent Dirichlet forms in the case of the homogeneous Boltzmann equation and the underlying process, see [5]}.
